section 4.1
Systems interacting via purely repulsive hard potentials are of interest because their
properties have a trivial dependence on temperature, they are
theoretically tractable, and they capture the qualitative effects of
intermolecular repulsion on material properties. These models are
particularly good at characterizing the structural features of
condensed phases [2]. While they are incapable of
exhibiting vapor-liquid transitions, properly formulated models have
been shown to exhibit a rich array of order-disorder transitions,
including freezing [56,39], polymorphism (in
hard-sphere mixtures) [73,74,75], and liquid crystalline
phases [76,26,77](seen, for example, in hard
spherocylinders).
As already mentioned in chapter 1, the hard sphere model gives a very good description of colloidal
dispersions that consist of non-charged spherical particles interacting via a
steep steric repulsion [14]. In particular, such colloidal
systems, if sufficiently monodisperse in size, are known to
crystallize at densities very close to that predicted by a hard-sphere
model [8]. However, colloidal particles frequently
exhibit considerable size polydispersity, depending on the way they
are synthesized. This polydispersity will affect the thermodynamic
properties, including the location and existence of any phase
transitions.
The influence of polydispersity on the solid-fluid transition in
colloidal suspensions was first examined by Dickinson and Parker
[78,79,80]. Their system consisted of
particles interacting via a screened Coulombic repulsion with a van
der Waals attractive term. They showed via molecular simulation
that the osmotic pressure of this system varies significantly with
size polydispersity. They estimated the solid-fluid coexistence properties
as a function of the polydispersity, but without attempting any sort of
rigorous free energy calculations or considering the possibility of size
fractionation (i.e., they assumed that the particle diameter distributions
in the coexisting solid and fluid phases are identical). They found that the
fluid-solid coexistence region narrows as the polydispersity increases, and they
surmised that the transition disappears entirely at sufficiently high
polydispersity. This value of the polydispersity has been called by
Dickinson and Parker the ``critical polydispersity''; this choice is
unfortunate as the phenomenon does not likely represent a continuous transition
because the solid and fluid phases have different symmetries. To
avoid any confusion with critical points at they are customarily
understood, we will instead refer to this polydispersity as the
``terminal polydispersity''.
The terminal polydispersity has come to be the subject of considerable
interest. For a triangular distribution, Dickinson and Parker
extrapolated the change in volume upon melting to zero, and estimated
the terminal polydispersity at 11% (we define the polydispersity as
the standard deviation of the particle size distribution, divided by the
mean). Later, Pusey [81] proposed a simple criterion for
the terminal polydispersity based on an analogy of the Lindemann
melting criterion; he also obtained a value of about 11%. Pusey
[8] performed experiments in which he observed that
dispersions with a polydispersity of 7.5% would (partly) freeze, while those
with a polydispersity of 12% did not.
Several authors applied density functional theory (DFT) to obtain the
phase diagram of polydisperse hard spheres [82,83].
These theories are significantly more sophisticated than those employed
in prior studies, and they give more attention to free energy criteria
in calculating the coexistence curves. They nevertheless do not
represent a completely rigorous treatment, as they too do not consider
the effect of fractionation on the coexistence properties. McRae and
Haymet [83] refer to this approximation as a ``constrained
eutectic''. The constrained eutectic is certainly valid at small polydispersity,
but it likely breaks down as the distribution of diameters becomes wide.
The DFT studies predict the terminal polydispersity at about 5-6%.
The purpose of the present work is to determine rigorously the phase diagram of
polydisperse hard spheres by molecular simulation, establish the
terminal polydispersity and test the validity of the constrained
eutectic assumption made explicit in ref. [83]. To do this properly
we must simulate a system having a truly continuous distribution of
particle sizes, rather than a many-component but nevertheless discrete
distribution. Such a simulation can be realized in the so-called
semigrand ensemble, which has the added advantage that it is well
suited for calculation of multicomponent phase equilibria. The
semigrand ensemble is explained in section 4.2. Section 4.3 describes
how the recently developed Gibbs-Duhem integration method [84]
can be applied to efficiently obtain the phase diagram of polydisperse
hard spheres by integration along the coexistence line. To start the
integration one needs the slope of the coexistence line in the monodisperse
limit; the means by which this is obtained is described in section 4.4.
In section 4.5, it is shown how scaling properties of the system can be
applied to greatly improve the accuracy of the simulations. The
simulation results are discussed in section 4.6. The existence of a
terminal polydispersity raises questions of continuity of the fluid
and solid states; we show in section 4.7 how this issue may be
resolved. Concluding remarks are presented in section 4.8
section 4.1